D₃×C₇×(C₁₁⋊D₁₁)⋊C₃ #44479086

order	30492
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₇ · Schur multiplier (H₂) = C₂×C₁₁

Aliases

direct product	D₃×C₇×(C₁₁⋊D₁₁)⋊C₃
unambiguous semidirect product	(D₃×C₁₁⋊D₁₁)⋊(C₃×C₇)

Splittings

Fitting, socle	C₃×C₇×C₁₁² ⋊ C₂²×C₃
derived	C₃×C₁₁² ⋊ C₂²×C₃×C₇
Hall	C₃×C₇×C₁₁²⋊C₃ ⋊ C₂²
Hall	C₃×C₁₁²⋊C₃ ⋊ C₂²×C₇
Hall	C₇×C₁₁² ⋊ C₂×C₃×D₃
Sylow	C₁₁² ⋊ C₂×C₃×D₃×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₃×(C₁₁⋊D₁₁)⋊C₃
commutator, cosocle, hypocenter	C₃×C₁₁²	C₂²×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₁²	C₂²×C₃
LCS	G ▷ C₃×C₁₁²
UCS	C₁ ◁ C₇

168 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 693

A minimal faithful permutation representation — 2 generators on 43 points:

(1,2,3)(4,5,6,7,8,9,10)(11,12,14)(13,27,28,17,18,39)(15,36,38,22,23,43)(16,21,41,25,26,35)(19,37,40,30,31,34)(20,42,24,33,32,29);(2,3)(11,13,19,32,43,41,35,38,42,30,17)(12,15,24,27,25,31,37,16,18,29,22)(14,20,21,34,28,36,23,39,40,26,33)

Conjugacy classes & character table

546 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 309);